This study aims to evaluate the process of neonatal phenylketonuria (PKU) screening from birth to admission to the pediatric metabolism polyclinic, determining delays in the screening program and the factors influencing them. This study was conducted during 2021–2023. Blood collection dates, results, and probable parameters causing delays in the screening program were recorded. This study included 118 infants. Admission time to the polyclinic was (mean ± SD) 25.2 ± 12.6 days (min–max: 3.4–78.9 days). Admission time was significantly high for refugees, those whose parents were consanguineous, and those who had more heel-prick blood samples taken (p < 0.001, p = 0.005, and p < 0.001, respectively). The first heel-prick blood phenylalanine (phe) level was significantly negatively correlated with the admission time (p < 0.001). Patients’ admission time whose first blood phe level < 240 μmol/L was statistically significantly higher than in those with ≥240 μmol/L (p < 0.001). We determined that there were delays in PKU screening from birth to admission to the polyclinic. Being a refugee, the presence of consanguineous marriages, the increase in the number of heel-prick tests, and blood phe levels at a range of 120–240 μmol/L were the factors that played a role in this delay. Taking steps to reduce the impact of these parameters can prevent delays in newborn PKU screening and increase the success of the screening program.